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Physica Scripta

Coherent-state path integral for a transition amplitude: A theory and applications

E. A. Kochetov, Vladimir Sergeevich Yarunin

Published Jan 1, 1995• 13 citations
Publisher

Abstract

The analytical methods of path integral (PI) calculations are presented for transition matrix elements in a basis of the generalized coherent states (CS'). Appropriate transformations in phase space are found and the stationary-phase method is developed with a particular emphasis on the PI boundary terms. Examples demonstrate the application of the general scheme for spin systems and the tests of the accuracy of the semiclassical approximation in nonlinear dynamics.

DOI
10.1088/0031-8949/51/1/007

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arXiv:1207.0012•Physical Review A•Jul 8, 2013•2 citations
The analytical methods of path integral (PI) calculations are presented for transition matrix elements in a basis of the generalized coherent states (CS'). Appropriate transformations in phase space are found and the stationary-phase method is developed with a particular emphasis on the PI boundary terms. Examples demonstrate the application of the general scheme for spin systems and the tests of the accuracy of the semiclassical approximation in nonlinear dynamics.
Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.
The SU(2) coherent-state path integral is used to represent the matrix element of a propagator in the SU(2) coherent-state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary-phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.
The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.
In applying the stationary phase approximation to coherent state path integrals a difficulty occurs; there are no classical paths that satisfy the boundary conditions of the path integral. Others have gotten around this problem by reevaluating the action. In this work it is shown that it is not necessary to reevaluate the action because the stationary phase approximation is applicable so long as the path, about which the expansion is performed, satisfies the associated Lagrange's equations of motion. It is not necessary for this path to satisfy the boundary conditions in order to apply the stationary phase approximation.
In this work, we derived a semiclassical approximation for the matrix elements of a quantum propagator in coherent states (CS) basis that avoids complex trajectories; it only involves real ones. For that purpose, we used the symplectically invariant semiclassical Weyl propagator obtained by performing a stationary phase approximation (SPA) for the path integral in the Weyl representation. After that, for the transformation to CS representation SPA is avoided; instead a quadratic expansion of the complex exponent is used. This procedure also allows us to express the semiclassical CS propagator uniquely in terms of the classical evolution of the initial point without the need of any root search typical of van Vleck--Gutzwiller-based propagators. For the case of chaotic Hamiltonian systems, the explicit time dependence of the CS propagator has been obtained. The comparison with a realistic chaotic system that derives from a quadratic Hamiltonian, the cat map, reveals that the expression here derived is exact up to quadratic Hamiltonian systems.